A transformation matrix is given for reducing a random sample of sizento a smaller set ofmuncorrelated variables, with the same mean and minimum common variance. This transformation may be applied to the classic Behrens-Fisher problem to obtain a solution similar to Scheffé's. An indication of the loss of information is also given. The transformation is also used to obtain an exact test of whether two regression planes are parallel, when the variances and sample sizes are unequal. Also a test of whether two regression planes are identical, when they are assumed parallel, can be derived easily by this method, and the results are similar to a recent solution by Potthoff. Some possible applications of the transformation to randomization tests concerning means is also given, the idea being to reduce the sample so that the amount of computations required would be feasible.